SVD algorithms to approximate spectra of dynamical systems

Dynamical indicators for the prediction of bursting phenomena in high-dimensional systems

“…As a result, many numerical techniques have been introduced to compute the flow invariant subspace in a numerically robust fashion (see, e.g., Greene and Kim [31], Dieci and Elia [32]). …”

Detecting exponential dichotomy on the real line: SVD and QR algorithms

“…, n; see (SVD-3) below. We now recall some results from [9,10,12] which will be needed to understand our techniques. Although the results in these cited works are given for real-valued A, they easily extend to the complex case; some details will be given below as needed.…”

“…A continuous SVD algorithm for the approximation of ± ED , together with the associated directions, have been developed in [9,10], to which we refer for details. Here we just summarize the main steps.…”

“…Suppose assumptions(10) and(16) hold and let > 0 be given. If ρ := max i,j ρ ij is sufficiently small, then there exists a time T > 0 such that Z c (T )− Z e (∞) ≤ where Z c (T ) is the computed Z at time T and Z e (∞) is the exact, limiting Z.…”

The singular value decomposition to approximate spectra of dynamical systems. Theoretical aspects

“…The first, and most relevant from the practical point of view, is that SVD techniques are harder to implement than QR techniques. We refer to [8] for a discussion of the delicate computational issues one has to face with SVD techniques, and for a particular implementation. The other is that, at least so far, we are only able to fully justify SVD techniques in case the original system has stable and distinct LEs, and this is true also in case we want to obtain Σ ED .…”